A note on Jeśmanowicz' conjecture concerning Pythagorean triples

Abstract

Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple. In this paper we give certain conditions for the equation (an)x + (bn)y = (cn)z to have positive integer solutions (x, y, z) with (x, y, z) ≠ (2, 2, 2). In particular, we show that x, y and z must be distinct.

A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples

“…But, the combination of (16) and (17) (3) has no exceptional solutions (x, y, z, n) with max{x, y} > min{x, y} > z. Thus, by the result of [9] and Theorem 1, the theorem is proved.…”

“…Hence, we see from (22) that z is even. Further, if 2 α || z, where α is a positive integer, by the lemma of [9] we have…”

“…In 1999, M.H. Le [9] proved that if (x, y, z, n) is an exceptional solution of (3), then one of the following conditions is satisfied:…”

“…However, the case x = y > z is not completely handled, the arguments used in [9] do not work in this case and need to be replaced by a different (actually simpler) argument. In this paper we shall complete the proof as follows:…”

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

“…In the final section, we prove the theorem as follows. First, assuming that there exists a non-trivial positive solution of equation (1.2) (i.e., (x, y, z) = (2, 2, 2)), we can divide our proof into two parts by using some necessary conditions on the existence of such a solution given by Le [6] and Deng [2]. For each case, we make use of some detailed local arguments using certain moduli, and do some calculations on the 2-adic valuation.…”